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I was surprised to find out that the continued fraction expansion of $e$ is $[2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,...]$. Even though $e$ is transcendental, this expansion is very nice.

We can describe it fairly succinctly by saying $a_0=2, a_1=1$ and for $n \geq 2$:

If $n \equiv 0$ or $n \equiv 1 \mod 3$ then $a_n=1.$

If $n \equiv 2 \mod 3$, $n+1=3k$, then $a_n = 2k$.

Now my question is: how can we characterize those numbers which have a continued fraction expansion that cannot be described by a finite formula?

By this I mean a finite number of statements which, taken together, describe every $a_n$.

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    $\begingroup$ In a way we are allowed to say that $e$ is trascendental because its continued fraction is really nice. In particular the convergents are related to the integrals $\int_{0}^{1} x^n(1-x)^n e^{-x}\,dx$ which have both a simple asymptotic behaviour and a very nice arithmetic structure, see the section about continued fractions in my notes. $\endgroup$
    – Jack D'Aurizio
    Apr 25, 2018 at 21:47

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I very much doubt that there is a satisfying characterization of numbers whose continued fractions can be described that way (you ask about "cannot"). You can argue that there are only countably many such numbers since there are only countably many finite "formulas". All the rationals are there, but so are some irrationals, like all the square roots, and some transcendentals, like $e$.

I don't know if all the algebraic numbers are there. They may well be.

It's not clear (to me) whether the set of those numbers has reasonable algebraic properties. See Are there simple algebraic operations for continued fractions? .

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